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Density Functional Theory

‣Probability of finding electron 1 in dx1 , electron 2 in dx2 etc‣Probability of finding electron 1 in volume element dr1 (other electrons can be anywhere)‣Probability of finding any electron in dr1

‣Integrate over spin-space coordinates of electrons 2-N and spin of 1:‣ρ(r) is the electron density

‣Wave function: a complicated function of 4 x Nel

variables. Not a physical "object". Wave function based methods scale poorly with system size and have high requirements on basis sets to properly describe the electron-electron cusp. ‣Density: a much simpler function of three variables. Experimental observable.

If we can use it directly, we might be able to come up with a simpler theory...Isovalue surfaces (electrons per unit volume)DFT: a quantum mechanical theory where the density is the central quantity

The density has cusps at the positions of the nuclei.We can fully define the Hamiltonian of the system simply by examining the density!By integrating the density we get the total number of electrons.

The shape of the cusps is directly related to the atomic number Z. DFTIn principle the ground state density contains everything there is to know. What is needed to make the connection (conceptually and practically)?

A function f(x) maps a number to another number. A functional F[f] takes a function as input and returns a number. We need a way to go from the density ρ(r) to the energy E,

i.e. we need to express the energy as a functional of the density, E[ρ]HamiltonianOwing to the Born-Oppenheimer approximation we perform a quantum calculation only on electrons; nuclei are "external" fixed objects which exert their potential to the electrons.External potential

1st HK theorem: a universal density functional exists The electron density determines the external potential. There is a one-to-one correspondence between densities and external potentials

(different external potentials always correspond to different densities) z the electronic energy can be expressed completely as a functional of the density2nd HK theorem: variational principle for the density only the exact ground-state density ρ(r) of H minimizes the value of its ground-state energy functional

z One can already use all of the above to do calculations! Minimize the energy with respect to var. density by constraining

Evaluation of kinetic energy has been a central problem. Assume a fictitious system of independent (non-interacting) electrons that have precisely the same density ρ(r) as the real physical system. Non-interacting system: decoupled coordinates, separable Hamiltonian. z Slater determinant: we are back to calculating orbitals!

Write F asBecauseit follows thatz guess EXC

functionals, use them in the Kohn-Sham orbital optimization procedurez EXC is the magic ingredient that corrects other errors, self-interaction, correlation... V.

Depends only on the scalar value of the density. Assumes that the exchange-correlation energy at every position in space for the molecule is the same as it would be for the uniform electron gas (UEG) having the same density as found at that position.Kinetic energy directly derived, similarly straightforward expression for exchange.

Several different expressions for the correlation energy: VWN (Vosko, Wilk, and Nusair), PW (Perdew and Wang), ... LDA is exact only for constant-density systems, yet it is already more accurate than Hartree-Fock as a general electronic structure theory.

Introduces additional dependence on the gradient of the density at a given point ("non-local functionals") Most GGA functionals are created as "add-ons" to LDA.

And this is where the fun begins!Do not contain empirical parameters: (focus on satisfying constraints and norms)Do contain empirical parameters: (focus on reproducing known quantities)Exchange: B86, PBE, ... | Correlation: PW91 Exchange: B, CAM, O, PW, mPW, X, ... | Correlation: B88, P86, LYPThe historical tension between the "first principles" and the fitting approaches continues to this day and shows no signs of resolution!GGA functionals first enabled real computational chemistry to be done! Still an excellent choice for many tasks in quantum chemistry. BP86 usually fine for geometry optimizations!2 Schools

of DFT

Try to achieve further improvement by including dependence on the Laplacian of the density (in practice: kinetic energy density).

Examples: B95, B98, TPSS, VSXC, M06-L, SCAN Usually limited improvement compared to GGAs.

Also known as Adiabatic Connection Method (ACM) functionals.They include fractions of exact Hartree-Fock exchange energy, calculated as a functional of the Kohn-Sham MOs. Controlled error cancellation.Most famous example: B3LYP (20% HF exchange).

Probably Hundreds of other functionals in this part of the DFT supermarket! In combination with meta-GGAs: hybrid meta-GGA functionals, e.g. TPSSh (10% HF). Range-separated functionals, variable exchange, e.g. CAM-B3LYP, ωB97. Often superior for spin state problems and spectroscopic properties.HF exchange as adjustable parameter: a blessing or a curse?•Low-spin/high-spin energy splittings for Fe complexes

•Cu-O2 adducts •Reaction barriers •Valence isomerism and bonding in transition metal clusters

B3LYPPBE0TPSShM06-LM06M06-2XBPHow can we decide?

They mix standard DFT exchange and correlation with HF exchange and an additional second-order perturbation theory contribution.

The PT2 contribution is obtained through a Møller-Plesset perturbational term (MP2) based on Kohn-Sham orbitals that were self-consistently optimized with respect to the first three terms.

Archetypal example: B2PLYP. Current "best": PWPB95Considered among the best functionals available ("top-rung")Victims of even more refitting and purpose-focused optimization (SCS-MP2, SOS-MP2) ... for kinetics, for thermochemistry, for ...

BUT not without important exceptions and unexpected failures

Octane vs. iso-octane: DFT underestimates the stability of branched hydrocarbon isomers e.g. B3LYP -9 kcal/mol vs exp. +2 kcal/mol

Origin: deficiency in medium-range correlation - DFT is too "short-sighted"Semi-empirical fix in the form of an add-on term: D3BJ, D4 (Grimme).Similar deficiency in any systems where dispersion is important.

Beneficial in the vast majority of cases, but always be careful...A more important question: relative energies are wrong - are spectroscopic properties also "wrong"?

* Several dispersion corrections* Additionally: Libxc functionals* Grids important in DFT (m)GGA - Hybrid - Double Hybrid

sequence generally true but depends on system and property Main problem: inconsistency, unpredictability. Literature is crucial.

Benchmarking is important. But: How to use it? How to do it? How not to get lost in it?You cannot get an easy answer for the "best" functional. Too many parameters! It is essential to understand how they interrelate in your case.Many types of system or properties are absent or under-represented in standard benchmark sets. E.g. most evaluation studies cannot sufficiently address the enormous chemical space of transition metal systems, heavier elements, ...System - Property - MethodWhat can I calculate with DFT? What functional should I choose?How to choose a method without relying on intentional bias & error cancellation?

GGA functionals adequate, sometimes better than hybrids. BP86 a decent choice, perhaps TPSS. Include dispersion corrections. Use with RI approximation (fast!)Geometries(m)GGA - Hybrid - Double Hybrid sequence generally true, but quality of results very system-dependent. Frequencies: often GGA sufficient Optical: TD-DFT... EPR: advantage of hybrid functionals for g, ZFS, hyperfines;

* Double hybrids discouraged for exchange coupling interactions!Energies:

Basis Sets

The SCF procedure involves solving single-electron equations for molecular orbitals.Can we think of a general, transferable, computer-friendly approach?Is there a way to standardize this task and make it transferable? We will express the molecular orbitals as linear combinations of atomic orbitals. Each atom comes with its set of AOs. We can construct any molecule we want and express any MO of this molecule in terms of the standard AOs of its constituent atoms.We need a standard set of building blocks, so that we don't have to guess or search for possible mathematical forms of MOs.CHEM6085 Density Functional Theory

4 Example: find the AOs from which the MOs of the following molecules will be built

The Hartree-Fock Roothaan MethodIt is difficult to vary the orbitals themselves. Rather what one does is to expand the orbitals in another set of auxiliary functions, the "basis set"

i (x)=c µi (x)

If the basis set {φ} would be mathematically "complete", the expansion would be exact. In practice, we have to live with less than complete basis set expansions. Carrying out the variation now with respect to the unknown "MO coefficients" c leads to the famous Hartree-Fock Roothaan equations. The MO coefficients must satisfy the following coupled set of nonlinear equations:

F (c)c µi i c νi S (allµ,i) Fψ i i i i =OrbitalEnergyofOrbitali

F=FockOperator

S=OverlapMatrix

Let's generalize this: Use more than one function for an "atomic orbital" (more flexible representation of MOs) Use mathematical forms that are convenient for calculations

(if they are not all that good, compensate by higher number)We call these more "general atomic orbitals" basis functionsWe assign a set of fixed functions (a basis set) to each atom. Then the task of finding the MOs is reduced to optimizing the MO expansion coefficients in terms of these fixed basis functions.The Hartree-Fock Roothaan MethodIt is difficult to vary the orbitals themselves. Rather what one does is to expand the orbitals in another set of auxiliary functions, the "basis set"

i (x)=c µi (x)

If the basis set {φ} would be mathematically "complete", the expansion would be exact. In practice, we have to live with less than complete basis set expansions. Carrying out the variation now with respect to the unknown "MO coefficients" c leads to the famous Hartree-Fock Roothaan equations. The MO coefficients must satisfy the following coupled set of nonlinear equations:

F (c)c µi i c νi S (allµ,i) Fψ i i i i =OrbitalEnergyofOrbitali

F=FockOperator

S=OverlapMatrix

Slater-type functions (exponent contains -r) are great because they best resemble hydrogen AOs and have the right shape close to the nucleus (cusp) and far from the nucleus (rate of decay)But it is computationally simpler to use Gaussian-type functions (exponent -r2

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